In this section we construct the weak solution for (REF ) with \(\kappa =1\) , \(\nu =0\) . We begin by introducing a mollified Boussinesq system constructed as in [1]}. We first recall the definition of the “retarded mollifier". Let \(\psi (x,t)\in C^\infty (\mathbb {R}^3\times \mathbb {R})\)  such that
\(\psi \ge 0,\quad \int _0^\infty \int \psi dxdt=1,\quad {\rm supp} \psi \subset \lbrace (x,t): |x|^2<t, 1<t<2\rbrace .\) 
